Natural automorphisms of the Douady space of points on a surface (Q2880075)

In the paper under review, the author presents the results of his investigations of the automorphisms of the Douady space of a complex analytic surface. In particular, the focus of the work is on the automorphisms, called \(natural\), induced by those of the surface.\newline Let \(n\) be a nonnegative integer and let \(S\) be a connected complex analytic surface, also assumed to be smooth and compact. The Douady space \(S^{[n]}\) parametrizes length \(n\), \(0\)-dimensional analytic subspaces of \(S\), and it reduces to the corresponding Hilbert scheme when \(S\) is algebraic.\newline In the first part of the paper, the author provides a direct generalization in the above setting of a result of \textit{L. Göttsche} [Math. Ann. 286, No. 1--3, 193--207 (1990; Zbl 0679.14007)], concerning the Hodge numbers of the usual Hilbert schemes. Such a generalization yields NEWLINE\[NEWLINE \dim \mathrm{Aut}(S^{[n]})=\dim \mathrm{Aut}(S), NEWLINE\]NEWLINE for any \(n\geq 1\) and any compact analytic surface \(S\). In particular, taking the lead from \textit{A. Beauville} [Classification of algebraic and analytic manifolds, Proc. Symp., Katata/Jap. 1982, Prog. Math. 39, 1--26 (1983; Zbl 0537.53057)], the author shows that for \(S\) simply connected, with trivial Picard group and such that \(h^0(S,T_S)=0\), one has NEWLINE\[NEWLINE\mathrm{Aut}(S^{[n]})\cong \mathrm{Aut}(S), NEWLINE\]NEWLINE for all \(n\geq 1\). From this, he then concludes that all the automorphisms of the Douady space of a generic \(K3\) surface are natural.\newline Motivated by the results summarized above, in the second part of his work the author shifts his attention back to the Douady space of any complex analytic surface. He carries out a study of the action of the natural automorphisms on the cohomology and, after some deformation theoretic preparation, he is also able to classify the fixed points of such automorphisms.